Embedded newform 124.4.p.a.3.12

• Label: 124.4.p.a.3.12
• Level: $124$
• Weight: $4$
• Character: 124.3
• Analytic conductor: $7.316$
• Analytic rank: $0$
• Dimension: $368$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 124 = 2^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	124.p (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.31623684071\)
Analytic rank:	\(0\)
Dimension:	\(368\)
Relative dimension:	\(46\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			3.12
Character	\(\chi\)	\(=\)	124.3
Dual form			124.4.p.a.83.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.25611 - 1.70586i) q^{2} +(-0.718973 + 6.84057i) q^{3} +(2.18011 + 7.69722i) q^{4} +(-2.50613 + 4.34075i) q^{5} +(13.2911 - 14.2066i) q^{6} +(4.92655 + 23.1776i) q^{7} +(8.21177 - 21.0848i) q^{8} +(-19.8664 - 4.22274i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 6 q^{2} + 6 q^{4} - 8 q^{5} - 9 q^{6} - 57 q^{8} + 360 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/124\mathbb{Z}\right)^\times\).

\(n\)	\(63\)	\(65\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{30}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.25611	−	1.70586i	−0.797657	−	0.603111i
							\(3\)	−0.718973	+	6.84057i	−0.138366	+	1.31647i	0.676338	+	0.736591i	\(0.263566\pi\)
−0.814704	+	0.579876i	\(0.803101\pi\)
\(5\)	−2.50613	+	4.34075i	−0.224155	+	0.388248i	−0.956066	−	0.293152i	\(-0.905296\pi\)
							0.731910	+	0.681401i	\(0.238629\pi\)
\(7\)	4.92655	+	23.1776i	0.266009	+	1.25147i	0.884820	+	0.465933i	\(0.154281\pi\)
							−0.618811	+	0.785540i	\(0.712386\pi\)
\(11\)	10.7036	+	11.8876i	0.293388	+	0.325841i	0.871760	−	0.489932i	\(-0.162979\pi\)
							−0.578372	+	0.815773i	\(0.696312\pi\)
\(13\)	−6.52626	−	14.6582i	−0.139235	−	0.312728i	0.830447	−	0.557098i	\(-0.188085\pi\)
							−0.969682	+	0.244371i	\(0.921419\pi\)
\(17\)	−87.1018	−	78.4268i	−1.24266	−	1.11890i	−0.988417	−	0.151760i	\(-0.951506\pi\)
							−0.254247	−	0.967139i	\(-0.581828\pi\)
\(19\)	−22.0023	+	49.4180i	−0.265667	+	0.596698i	−0.996288	−	0.0860854i	\(-0.972564\pi\)
							0.730621	+	0.682784i	\(0.239231\pi\)
\(23\)	6.23834	+	19.1996i	0.0565558	+	0.174061i	0.975344	−	0.220690i	\(-0.0708309\pi\)
							−0.918788	+	0.394751i	\(0.870831\pi\)
\(29\)	2.41871	+	3.32907i	0.0154877	+	0.0213170i	0.816691	−	0.577076i	\(-0.195806\pi\)
							−0.801203	+	0.598393i	\(0.795806\pi\)
\(31\)	−64.3485	−	160.157i	−0.372817	−	0.927905i
							\(37\)	−58.9367	+	34.0271i	−0.261869	+	0.151190i	−0.625187	−	0.780475i	\(-0.714977\pi\)
0.363318	+	0.931665i	\(0.381644\pi\)
\(41\)	2.65651	+	25.2750i	0.0101189	+	0.0962753i	0.998416	−	0.0562683i	\(-0.0179202\pi\)
							−0.988297	+	0.152544i	\(0.951254\pi\)
\(43\)	−294.080	−	130.933i	−1.04295	−	0.464351i	−0.187515	−	0.982262i	\(-0.560043\pi\)
							−0.855435	+	0.517911i	\(0.826710\pi\)
\(47\)	142.397	−	195.993i	0.441930	−	0.608265i	−0.528709	−	0.848803i	\(-0.677324\pi\)
							0.970640	+	0.240538i	\(0.0773239\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.4.p.a.3.12	✓	368
4.3	odd	2	inner	124.4.p.a.3.30	yes	368
31.21	odd	30	inner	124.4.p.a.83.30	yes	368
124.83	even	30	inner	124.4.p.a.83.12	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.4.p.a.3.12	✓	368	1.1	even	1	trivial
124.4.p.a.3.30	yes	368	4.3	odd	2	inner
124.4.p.a.83.12	yes	368	124.83	even	30	inner
124.4.p.a.83.30	yes	368	31.21	odd	30	inner